Abstract
Chen and Flum showed that any FPT-approximation of the k-Clique problem is not in para- AC 0 and the k-DominatingSet (k-DomSet) problem could not be computed by para- AC 0 circuits. It is natural to ask whether the f ( k ) -approximation of the k-DomSet problem is in para- AC 0 for some computable function f. Very recently it was proved that assuming W [ 1 ] ≠ FPT , the k-DomSet problem cannot be f ( k ) -approximated by FPT algorithms for any computable function f by S., Laekhanukit and Manurangsi and Lin, seperately. We observe that the constructions used in Lin’s work can be carried out using constant-depth circuits, and thus we prove that para- AC 0 circuits could not approximate this problem with ratio f ( k ) for any computable function f. Moreover, under the hypothesis that the 3-CNF-SAT problem cannot be computed by constant-depth circuits of size 2 ε n for some ε > 0 , we show that constant-depth circuits of size n o ( k ) cannot distinguish graphs whose dominating numbers are either ≤k or > log n 3 log log n 1 / k . However, we find that the hypothesis may be hard to settle by showing that it implies NP ⊈ NC 1 .
Highlights
The dominating set problem is often regarded as one of the most important NP-complete problems in computational complexity
G = (V, E) and a number k ∈ N, to decide the minimum dominating set of G has a size of at most k. This problem is tightly connected to the set cover problem, which was firstly shown to be NP-complete in Karp’s famous NP-completeness paper [1]
The set cover conjecture asserts that for every fixed ε > 0, no algorithm can solve the set cover problem in time 2(1−ε)n poly(m), even if set sizes are bounded by ∆ = ∆(ε) [2,3]
Summary
The dominating set problem is often regarded as one of the most important NP-complete problems in computational complexity. The size of the minimum dominating set is at most k, The size of the minimum dominating set is greater than Note that this theorem implies the nonexistence of para-AC0 circuits which f (k)-approximates the k-D OM S ET problem for any computable function f. This is because if there is an f (k )-approximation para-AC0 circuit Cn,k whose size is g(k)poly(n), we can construct a constant-depth para-AC0 circuit C0n,k log n to distinguish the size of the minimum dominating set is at most k or greater than log log n as log n follows. Compared with the conference version [30] of this article, the proofs of Lemmas 1–7 are firstly given here; some results are slightly improved by more careful analyses
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